In this subsection, we assume \(\)  is connected solvable \(k\) -algebraic group. Recall that by structure theory of connected solvable groups [1]}, \(\)  admits a Levi decomposition such that the Levi subgroup is a maximal torus defined over \(k\) .


Let \(\)  be a connected solvable \(k\) -algebraic group, and let \(\ltimes {u}{}\)  be a \(k\) -Levi decomposition of \(\) .
Denote by \(C=\lbrace t\in T : \forall u\in {u}{G},\ [t,u]=e\rbrace \)  the kernel of the conjugation action of \(T\)  on \({u}{G}\) .
Then there exist a finite dimensional \(k\) -vector space \(V\) , and an algebraic \(k\) -rational representation \(\pi \colon G\rightarrow (V)\) , with kernel \(\ker (\pi )=C\times {u}{G}\)  such that:
